8.05c Sections and contours: sketch and relate to surface

2 A surface has equation \(z = x y ^ { 2 } - 4 x ^ { 2 } y - 2 x ^ { 3 } + 27 x ^ { 2 } - 36 x + 20\).

1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
2. Find the coordinates of the four stationary points on the surface, showing that one of them is \(( 2,4,8 )\).
3. Sketch, on separate diagrams, the sections of the surface defined by \(x = 2\) and by \(y = 4\). Indicate the point \(( 2,4,8 )\) on these sections, and deduce that it is neither a maximum nor a minimum.
4. Show that there are just two points on the surface where the normal line is parallel to the vector \(36 \mathbf { i } + \mathbf { k }\), and find the coordinates of these points.

7 The 'parabolic' TV satellite dish in the diagram can be modelled by the surface generated by the rotation of part of a parabola around a vertical \(z\)-axis. The model is represented by part of the surface with equation \(z = \mathrm { f } ( x , y )\) and \(O\) is on the surface. The point \(P\) is on the rim of the dish and directly above the \(x\)-axis.
The object, \(B\), modelled as a point on the \(z\)-axis is the receiving box which collects the TV signals reflected by the dish. \includegraphics[max width=\textwidth, alt={}, center]{f2166e0a-cd4c-40af-b4b4-04ef4919d996-3_753_995_584_525}

1. The horizontal plane \(\Pi _ { 1 }\), containing the point \(P\), intersects the surface of the model in a contour of the surface.
   1. Sketch this contour in the Printed Answer Booklet.
   2. State a suitable equation for this contour.
   3. A second plane, \(\Pi _ { 2 }\), containing both \(P\) and the \(z\)-axis, intersects the surface of the model in a section of the surface.
      (a) Sketch this section in the Printed Answer Booklet.
      (b) State a suitable equation for this section.
   4. A proposed equation for the surface is \(z = a x ^ { 2 } + b y ^ { 2 }\). What can you say about the constants \(a\) and \(b\) within this equation? Justify your answers.
   5. The real TV satellite dish has the following measurements (in metres): the height of \(P\) above \(O\) is 0.065 and the perimeter of the rim is 2.652 . Using this information, calculate correct to three decimal places the values of

5 A research student is using 3-D graph-plotting software to model a chain of volcanic islands in the Pacific Ocean. These islands appear above sea-level at regular intervals, (approximately) distributed along a straight line. Each island takes the form of a single peak; also, along the line of islands, the heights of these peaks decrease in size in an (approximately) regular fashion (see Fig. 1.1). \begin{figure}[h] \end{figure} The student's model uses the surface with equation \(\mathrm { z } = \sin \mathrm { x } + \sin \mathrm { y }\), a part of which is shown in Fig. 1.2 below. The surface of the sea is taken to be the plane \(z = 0\). \begin{figure}[h] \end{figure}

1. - Describe two problems with this model.
   Explain what has happened.

8 A surface, \(C\), is given by the equation \(z = \mathrm { f } ( x , y )\) for all real values of \(x\) and \(y\). You are given that \(C\) has the following properties.

• The surface is continuous for all \(x\) and \(y\).
• The contour \(z = - 1\) is a single point on the \(z\)-axis.
• For \(- 1 < a < 1\), the contour \(z = a\) is a pair of circles with different radiuses but each having the same centre \(( 0,0 , a )\).
• The contour \(z = 1\) consists of the circle, centre \(( 0,0,1 )\) and radius 1 .

Sketch a possible section of \(C\) corresponding to \(y = 0\).

1 The surface \(S\) is defined for all real \(x\) and \(y\) by the equation \(z = x ^ { 2 } + 2 x y\). The intersection of \(S\) with the plane \(\Pi\) gives a section of the surface. On the axes provided in the Printed Answer Booklet, sketch this section when the equation of \(\Pi\) is each of the following.

1. \(x = 1\)
2. \(y = 1\)

2 The surface \(S\) is given by \(z = x ^ { 2 } + 4 x y\) for \(- 6 \leqslant x \leqslant 6\) and \(- 2 \leqslant y \leqslant 2\).

1. 1. Write down the equation of any one section of \(S\) which is parallel to the \(x\)-z plane
   2. Sketch the section of (a)(i) on the axes provided in the Printed Answer Booklet.
2. Write down the equation of any one contour of \(S\) which does not include the origin.

A surface \(S\) is defined by \(z = 4x^2 + 4y^2 - 4x + 8y + 11\).

1. Show that the point P\((0.5, -1, 6)\) is the only stationary point on \(S\). [2]
2. 1. On the axes in the Printed Answer Booklet, draw a sketch of the contour of the surface corresponding to \(z = 42\). [2]
   2. By using the sketch in part (b)(i), deduce that P must be a minimum point on \(S\). [3]
3. In the section of \(S\) corresponding to \(y = c\), the minimum value of \(z\) occurs at the point where \(x = a\) and \(z = 22\). Find all possible values of \(a\) and \(c\). [4]

In this question you must show detailed reasoning. A surface \(S\) is defined by \(z = \mathrm{f}(x, y)\) where \(\mathrm{f}(x, y) = x^3 + x^2 y - 2y^2\).

1. On the coordinate axes in the Printed Answer Booklet, sketch the section \(z = \mathrm{f}(2, y)\) giving the coordinates of any turning points and any points of intersection with the axes. [4]
2. Find the stationary points on \(S\). [7]